In the present work, we use a phase-field
approach based on the time-dependent Ginzburg-Landau equations [1]}, [2]}, [3]}, [4]}, [5]}.
This modelling approach is well established for ferroelectrics and
has been successfully applied to study defects [6]}, [7]},
domain structures in thin films [8]}, [9]}, the effect
of flexoelectric coupling on domain patterns [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]},
and microstructural evolution under electro-mechanical loading [19]}.
The phase-field approach has made a significant contribution in understanding
nanoscale ferroelectrics and reliably accounts for energy contributions
from polarization gradients and misfit strains at domain walls [20]}.
Several reviews provide a broader appreciation of the capabilities
of the method [9]}, [22]}, [23]}. The phase-field simulations
have also been employed to simulate macroscopic ferroelectric properties
by modelling periodic boundary conditions [24]}, [2]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}.
For example, systematic studies on the effect of electro-mechanical
boundary conditions on hysteresis, butterfly-loops and polarization
switching have been conducted using periodic conditions [2]}, [26]}, [27]}, [28]}, [29]}, [30]}.
While there exists a substantial literature making use of phase field
models to study periodic patterns with stripe-like features in both
2D [2]}, [26]}, [27]}, [28]}, [29]} and 3D [30]}, [44]}
simulations, to date there is relatively little work done to systematically
explore the stability of nanoscale periodic patterns with 2D checkerboard-like
features or with cylindrical domains in 3D that are predicted to form
in tetragonal ferroelectrics [45]}, [46]}, [47]}, which is
the goal of the present work. In this paper, the stability of these
complex nanoscale periodic patterns under externally applied strain
and electric field is also explored. 
